Module FormalML.lib_utils.LibUtilsLattice


Definition of a lattice, loosely based on ideas from "A reflection-based proof tactic for lattices in Coq" and http://www.pps.univ-paris-diderot.fr/~sozeau/repos/coq/order/

Require Import RelationClasses.
Require Import Morphisms.
Require Import Equivalence.
Require Import EquivDec.

Section Lattice.

  Definition part_le {A} {eqA} {R} `{part:PartialOrder A eqA R} : _ := R.
  Infix "≤" := part_le (at level 70, no associativity).

  Definition associative {A} eqA `{equivA : Equivalence A eqA} (op : A->A-> A)
    := forall x₁ x₂ x₃, equiv (op (op x₁ x₂) x₃) (op x₁ (op x₂ x₃)).

  Definition commutative {A} eqA `{equivA : Equivalence A eqA} (op : A->A->A)
    := forall x₁ x₂, equiv (op x₁ x₂) (op x₂ x₁).

  Definition idempotent {A} eqA `{equivA : Equivalence A eqA} (op : A->A->A)
    := forall x, equiv (op x x) x.

  Definition absorptive {A} eqA `{equivA : Equivalence A eqA}
             (op1 op2 : A->A->A)
    := forall x y, op1 x (op2 x y) === x.

    Definition is_unit_r {A} eqA `{equivA : Equivalence A eqA}
             (op : A->A->A) (unit:A)
    := forall a, op a unit === a.

    Definition is_unit_l {A} eqA `{equivA : Equivalence A eqA}
             (op : A->A->A) (unit:A)
    := forall a, op unit a === a.

  Class Lattice (A:Type) (eqA:A->A->Prop) {equivA:Equivalence eqA}
    := {
        meet : A -> A -> A;
        join : A -> A -> A;

        meet_morphism :> Proper (eqA ==> eqA ==> eqA) meet ;
        join_morphism :> Proper (eqA ==> eqA ==> eqA) join ;
        
        meet_commutative : commutative eqA meet;
        meet_associative : associative eqA meet;
        meet_idempotent : idempotent eqA meet;

        join_commutative : commutative eqA join;
        join_associative : associative eqA join;
        join_idempotent : idempotent eqA join;

        meet_absorptive : absorptive eqA meet join;
        join_absorptive : absorptive eqA join meet
      }.

  Infix "⊓" := meet (at level 70).
  Infix "⊔" := join (at level 70).

A lattice that is consistent with a specified partial order.

  Class OLattice {A:Type}
        (eqA:A->A->Prop)
        (ordA:A->A->Prop)
        {equivA:Equivalence eqA}
        {lattice:Lattice A eqA}
        {pre:PreOrder ordA}
        {po:PartialOrder eqA ordA}
    := {
        consistent_meet: forall a b, a ≤ b <-> a ⊓ b === a
        }.

  Section oprops.
    Context {A eqA ordA} `{olattice:OLattice A eqA ordA}.

    Lemma consistent_join
    : forall a b, a ≤ b <-> a ⊔ b === b.
  Proof.
    intros.
    rewrite consistent_meet.
    split; intros eqq.
    - rewrite <- eqq.
      rewrite join_commutative, meet_commutative.
      rewrite (join_absorptive b a).
      reflexivity.
    - rewrite <- eqq.
      rewrite (meet_absorptive a b).
      reflexivity.
  Qed.

  Lemma meet_leq_l a b: (a ⊓ b) ≤ a.
  Proof.
    rewrite consistent_meet.
    rewrite meet_commutative.
    rewrite <- meet_associative, meet_idempotent.
    reflexivity.
  Qed.

  Lemma meet_leq_r a b: (a ⊓ b) ≤ b.
  Proof.
    rewrite meet_commutative.
    apply meet_leq_l.
  Qed.

  Theorem meet_most {a b c} : a ≤ b -> a ≤ c -> a ≤ (b ⊓ c).
  Proof.
    intros sub1 sub2.
    rewrite consistent_meet in sub1,sub2.
    rewrite <- sub1, <- sub2.
    rewrite meet_associative.
    rewrite (meet_commutative c b).
    apply meet_leq_r.
  Qed.

  Lemma join_leq_l a b: a ≤ (a ⊔ b).
  Proof.
    rewrite consistent_join.
    rewrite <- join_associative, join_idempotent.
    reflexivity.
  Qed.

  Lemma join_leq_r a b: b ≤ (a ⊔ b).
  Proof.
    rewrite join_commutative.
    apply join_leq_l.
  Qed.

  Theorem join_least {a b c} : a ≤ c -> b ≤ c -> (a ⊔ b) ≤ c.
  Proof.
    intros sub1 sub2.
    rewrite consistent_join in sub1,sub2.
    rewrite consistent_join.
    rewrite join_associative.
    rewrite sub2, sub1.
    reflexivity.
  Qed.

    Global Instance meet_leq_proper :
    Proper (part_le ==> part_le ==> part_le) meet.
  Proof.
    unfold Proper, respectful.
    intros a1 a2 aleq b1 b2 bleq.
    rewrite consistent_meet in aleq, bleq |- *.
    rewrite meet_associative.
    rewrite (meet_commutative a2 b2).
    rewrite <- (meet_associative b1 b2 a2).
    rewrite bleq.
    rewrite (meet_commutative b1 a2).
    rewrite <- meet_associative.
    rewrite aleq.
    reflexivity.
  Qed.

  Global Instance join_leq_proper :
    Proper (part_le ==> part_le ==> part_le) join.
  Proof.
    unfold Proper, respectful.
    intros a1 a2 aleq b1 b2 bleq.
    rewrite consistent_join in aleq, bleq |- *.
    rewrite join_associative.
    rewrite (join_commutative a2 b2).
    rewrite <- (join_associative b1 b2 a2).
    rewrite bleq.
    rewrite (join_commutative b2 a2).
    rewrite <- join_associative.
    rewrite aleq.
    reflexivity.
  Qed.


If the equivalence relation is decidable, then we can decide the leq relation using either meet or join
  Lemma leq_dec_meet `{dec:EqDec A eqA} a b : {a ≤ b} + {~ a ≤ b}.
  Proof.
    generalize (consistent_meet a b).
    destruct (meet a b == a); unfold equiv, complement in *; intuition.
  Defined.

  Lemma leq_dec_join `{dec:EqDec A eqA} a b : {a ≤ b} + {~ a ≤ b}.
  Proof.
    generalize (consistent_join a b).
    destruct (join a b == b); unfold equiv, complement in *; intuition.
  Defined.

  End oprops.

  Class BLattice {A:Type}
        (eqA:A->A->Prop)
        {equivA:Equivalence eqA}
        {lattice:Lattice A eqA}
    := {
        top:A;
        bottom:A;
        join_bottom_r: is_unit_r eqA join bottom;
        meet_top_r: is_unit_r eqA meet top
        }.

  Section bprops.

    Lemma is_unit_l_r {A} eqA `{equivA : Equivalence A eqA}
          (op : A->A->A) : commutative eqA op ->
                           (forall unit, is_unit_l eqA op unit <-> is_unit_r eqA op unit).
    Proof.
      unfold is_unit_l, is_unit_r.
      intros comm unit.
      split; intros; rewrite comm; trivial.
    Qed.

    Context {A eqA} `{blattice:BLattice A eqA}.

    Lemma join_bottom_l: is_unit_l eqA join bottom.
    Proof.
      apply is_unit_l_r.
      - apply join_commutative.
      - apply join_bottom_r.
    Qed.

    Lemma meet_top_l: is_unit_l eqA meet top.
    Proof.
      apply is_unit_l_r.
      - apply meet_commutative.
      - apply meet_top_r.
    Qed.

    Lemma meet_bottom_r : forall a, a ⊓ bottom === bottom.
    Proof.
      intros a.
      generalize (join_bottom_l a); intros leq.
      rewrite <- leq.
      rewrite meet_commutative.
      rewrite (meet_absorptive bottom a).
      reflexivity.
    Qed.

    Lemma meet_bottom_l : forall a, bottom ⊓ a === bottom.
    Proof.
      intros a.
      rewrite meet_commutative, meet_bottom_r.
      reflexivity.
    Qed.

    Lemma join_top_r : forall a, a ⊔ top === top.
    Proof.
      intros a.
      generalize (meet_top_l a); intros leq.
      rewrite <- leq.
      rewrite join_commutative.
      rewrite (join_absorptive top a).
      reflexivity.
    Qed.

    Lemma join_top_l : forall a, top ⊔ a === top.
    Proof.
      intros a.
      rewrite join_commutative, join_top_r.
      reflexivity.
    Qed.

  End bprops.

  Section boprops.

    Context {A eqA ordA equiv}
            {lattice:@Lattice A eqA equiv}
            {pre part}
            {blattice:@BLattice A eqA equiv lattice}
            {olattice:@OLattice A eqA ordA equiv lattice pre part}.

    Lemma le_top : forall a, a ≤ top.
    Proof.
      intros.
      rewrite consistent_join, join_top_r.
      reflexivity.
    Qed.
    
    Lemma bottom_le : forall a, bottom ≤ a.
    Proof.
      intros.
      rewrite consistent_join, join_bottom_l.
      reflexivity.
    Qed.

    Lemma bottom_le_top : bottom ≤ top.
    Proof.
      apply le_top.
    Qed.
    
  End boprops.

End Lattice.

Infix "≤" := part_le (at level 70, no associativity).
Infix "⊓" := meet (at level 70).
Infix "⊔" := join (at level 70).

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